; Analysis of "Everyone_i thinks he_i left" let g = \f g x. f (g x) in let z = \f g x. f (g x) x in -let everyone = \P. FORALL x (P x) in let he = \x. x in -everyone ((z thinks) (g left he)) +let everyone = \P. FORALL x (P x) in + +everyone (z thinks (g left he)) ~~> FORALL x (thinks (left x) x)@@ -37,23 +36,22 @@ Several things to notice: First, pronouns denote identity functions. As Jeremy Kuhn has pointed out, this is related to the fact that in the mapping from the lambda calculus into combinatory logic that we discussed earlier in the course, bound variables translated to I, the -identity combinator. This is a point we'll return to in later -discussions. +identity combinator (see additional comments below). We'll return to +the idea of pronouns as identity functions in later discussions. Second, *g* plays the role of transmitting a binding dependency for an -embedded constituent to a containing constituent. If the sentence had -been *Everyone_i thinks Bill said he_i left*, there would be an -occurrence of *g* in the most deeply embedded clause (*he left*), and -another occurrence of (a variant of) *g* in the next most deeply -embedded clause (*Bill said he left*). - -Third, binding is accomplished by applying *z* not to the element that -will (in some pre-theoretic sense) bind the pronoun, here, *everyone*, -but by applying *z* instead to the predicate that will take *everyone* -as an argument, here, *thinks*. The basic recipe in Jacobson's system -is that you transmit the dependence of a pronoun upwards through the -tree using *g* until just before you are about to combine with the -binder, when you finish off with *z*. +embedded constituent to a containing constituent. + +Third, one of the peculiar aspects of Jacobson's system is that +binding is accomplished not by applying *z* to the element that will +(in some pre-theoretic sense) bind the pronoun, here, *everyone*, but +rather by applying *z* instead to the predicate that will take +*everyone* as an argument, here, *thinks*. + +The basic recipe in Jacobson's system, then, is that you transmit the +dependence of a pronoun upwards through the tree using *g* until just +before you are about to combine with the binder, when you finish off +with *z*. (There are examples with longer chains of *g*'s below.) Last week we saw a reader monad for tracking variable assignments: @@ -68,10 +66,10 @@ let shift (c : char) (v : int reader) (u : 'a reader) = let lookup (c : char) : int reader = fun (e : env) -> List.assoc c e;; -(We've used a simplified term for the bind function in order to +(We've used a simplified term for the bind function here in order to emphasize its similarities with Jacboson's geach combinator.) -This monad boxed up a value along with an assignment function, where +This monad boxes up a value along with an assignment function, where an assignemnt function was implemented as a list of `char * int`. The idea is that a list like `[('a', 2); ('b',5)]` associates the variable `'a'` with the value 2, and the variable `'b'` with the value 5. @@ -100,18 +98,10 @@ kind of value that can be linked into a structure is an individual of type `e`. It is easy to make the monad polymorphic in the type of the linked value, which will be necessary to handle, e.g., paycheck pronouns. -Note that in addition to `unit` being Curry's K combinator, this `ap` -is the S combinator. Not coincidentally, recall that the rule for -converting an arbitrary application `M N` into Combinatory Logic is `S -[M] [N]`, where `[M]` is the CL translation of `M` and `[N]` is the CL -translation of `N`. There, as here, the job of `ap` is to take an -argument and make it available for any pronouns (variables) in the two -components of the application. - In the standard reader monad, the environment is an assignment function. Here, instead this monad provides a single value. The idea -is that this is the value that will replace the pronouns linked to it -by the monad. +is that this is the value that will be used to replace the pronoun +linked to it by the monad. Jacobson's *g* combinator is exactly our `lift` operator: it takes a functor and lifts it into the monad. Surely this is more than a coincidence. @@ -136,14 +126,14 @@ the parallel with the reader monad even more by writing a `shift` operator that used `unit` to produce a monadic result, if we wanted to. The monad version of *Everyone_i thinks he_i left*, then (remembering -that `he = fun x -> x`, and that `l a f = f a`) is +that `he = fun x -> x`, and letting `t a f = f a`) is

everyone (z thinks (g left he)) ~~> forall w (thinks (left w) w) -everyone (z thinks (g (l bill) (g said (g left he)))) +everyone (z thinks (g (t bill) (g said (g left he)))) ~~> forall w (thinks (said (left w) bill) w)@@ -154,3 +144,96 @@ Jacobson's variable-free semantics is essentially a reader monad. One of Jacobson's main points survives: restricting the reader monad to a single-value environment eliminates the need for variable names. + +Binding more than one variable at a time +---------------------------------------- + +It requires some cleverness to use the link monad to bind more than +one variable at a time. Whereas in the standard reader monad a single +environment can record any number of variable assignments, because +Jacobson's monad only tracks a single dependency, binding more than +one pronoun requires layering the monad. (Jacobson provides some +special combinators, but we can make do with the ingredients already +defined.) + +Let's take the following sentence as our target, with the obvious +binding relationships: + +

+ John believes Mary said he thinks she's cute. + | | | | + | |---------|---------| + | | + |-----------------------| ++ +It will be convenient to +have a counterpart to the lift operation that combines a monadic +functor with a non-monadic argument: + +

+ let g f v = ap (unit f) v;; + let g2 u a = ap u (unit a);; ++ +As a first step, we'll bind "she" by "Mary": + +

+believes (z said (g2 (g thinks (g cute she)) she) mary) john + +~~> believes (said (thinks (cute mary) he) mary) john ++ +As usual, there is a trail of *g*'s leading from the pronoun up to the +*z*. Next, we build a trail from the other pronoun ("he") to its +binder ("John"). + +

+believes + said + thinks (cute she) he + Mary + John + +believes + z said + (g2 ((g thinks) (g cute she))) he + Mary + John + +z believes + (g2 (g (z said) + (g (g2 ((g thinks) (g cute she))) + he)) + Mary) + John ++ +In the first interation, we build a chain of *g*'s and *g2*'s from the +pronoun to be bound ("she") out to the level of the first argument of +*said*. + +In the second iteration, we treat the entire structure as ordinary +functions and arguments, without "seeing" the monadic region. Once +again, we build a chain of *g*'s and *g2*'s from the currently +targeted pronoun ("he") out to the level of the first argument of +*believes*. (The new *g*'s and *g2*'s are the three leftmost). + +

+z believes (g2 (g (z said) (g (g2 ((g thinks) (g cute she))) he)) mary) john + +~~> believes (said (thinks (cute mary) john) mary) john ++ +Obviously, we can repeat this strategy for any configuration of pronouns +and binders. + +This binding strategy is strongly reminiscent of the translation from +the lambda calculus into Combinatory Logic that we studied earlier +(see the lecture notes for week 2). Recall that bound pronouns ended +up translating to I, the identity combinator, just like Jacobson's +identity functions for pronouns; abstracts (\a.M) whose body M did not +contain any occurrences of the pronoun to be bound translated as KM, +just like our unit, which you will recognize as an instance of K; and +abstracts of the form (\a.MN) translated to S[\aM][\aN], just like our +ap rule.